As reported by Stuart Taylor way back in 1988, Justice Kennedy took no part in consideration of the landmark appointment/removal power case of Morrison v. Olson:

Justice Anthony M. Kennedy, who joined the Supreme Court last week, has disqualified himself in one of the most momentous cases the Court has faced in years, the challenge by the Reagan Administration and others to the constitutionality of the special prosecutor law. …

Justice Kennedy would not disclose the reason for his action, Toni House, the Court’s public information officer, said today. Justice Kennedy’s decision to recuse himself was reported today by The Wall Street Journal and The Washington Post. … Justice Kennedy is not known to have had any relationship with any of those involved in the Olson investigation.

I could not find a definitive answer to this question, but Alexia Morrison, the independent counsel at issue in the case–who also argued her own case–offered this observation:

While I do not know why Justice Kennedy recused himself — I understand he is one of those who does not give a reason, though some do — I can give you a bit of information about the timing issues raised by your inquiry. Our case was one considered at Kennedy’s first Friday conference — where the justices determine whether or not to grant cert — in January 1988. He recused himself at that time from all consideration of the case. The Court granted our petition for cert at that Friday conference, along with our request for expedited treatment, and announced that Kennedy would not sit on the case. So, we knew from the beginning that we would have only eight justices sitting — not a good situation for the party trying to overturn what was, for us, a dreadful decision at the circuit. Under our expedited schedule, the case was briefed and argued by the end of April and the decision announced at the Court’s last day of that session in June. Hope this helps. Cheers. Alexia

The case turned out to be 7-1, with only Justice Scalia dissenting.

Recall that Kennedy (as did Scalia) voted with the majority  in Free Enterprise Fund v. PCAOB, holding that removal of members of the PCAOB violated the separation of powers.

So how was Alexia Morrison appointed? Through the actions of Chief Justice Rehnquist, as he notes in Footnote 3:

The Special Division is a division of the United States Court of Appeals for the District of Columbia Circuit. 28 U.S.C. § 49 (1982 ed., Supp. V). The court consists of three circuit court judges or justices appointed by the Chief Justice of the United States. One of the judges must be a judge of the United States Court of Appeals for the District of Columbia Circuit, and no two of the judges may be named to the Special Division from a particular court. The judges are appointed for 2-year terms, with any vacancy being filled only for the remainder of the 2-year period. Ibid.

Rehnquist, indirectly, had something to do with the appointment of Morrison. I could not find the names of the 3 judges on the special division.